Question

Let a , b , c are roots of equation x(x-2)(x-4)=mx+6.
Find m such that (a+b)(b+c)(c+a)=60.

Answer 1

\[x(x-2)(x-4)=mx+6\]\[x^3-6x^2+8x=mx+6\]\[x^3-6x^2+(8-m)x-6=0\]\[a+b+c=6\quad,\quad ab+bc+ca=8-m\quad,\quad abc=6\]\[(a+b)(b+c)(c+a)=(6-c)(6-a)(6-b)=\]\[=(6-c)(36-6a-6b+ab)=\]\[=216-36(a+b+c)+6(ab+bc+ca)-abc=\]\[=216-36\cdot 6+6(8-m)-6=42-6m\]\[42-6m=60\quad\Rightarrow\quad m=-3\]